I'm strugging with a portion of Rudin's proof.(adsbygoogle = window.adsbygoogle || []).push({});

Quick statement of the bulk of the theorem:

Let X be a locally compact Hausdorff space. Let A be a positive linear functional on Cc(X) (continous functions with compact support). Then (among other things), there exists a measure u() that represents A:

A(f) = Integral(fdu) for every f in Cc(X).

Now assuming that, he shows that u(K) for any compact set K is finite.

He basically shows that u(K) <= A(f) for some f in Cc(X) with 0 <= f <= 1. This far I follow. But then he immediately concludes that therefore u(K) is finite for any compact K.

Is there some basic property of positive linear functionals that makes them always finite? There is a somewhat similar proof in Rudin's Principles of Mathematical Analysis where he shows that the norm of linear functionals on finite dimensional vector spaces is finite. But that proof assumes a finite dimensional space. So I can't see how to apply that proof here.

Any help is appreciated.

thanks

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# Riesz representation theorem

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